Determining polynomial upper and lower bounds of general
functions has been discussed in the literature
[45, 14, 18, 72]. In the results
cited, optimality is determined via the norm,
as done in section .

In [14],
it is shown that if
is bounded, and finite for at least n+1 points, then
there exists optimal lower and upper degree n
bounds. It is also shown that if g is continuous
on [0,1], and differentiable on (0,1), then
the optimal bounds are unique.
It is also established that for g with
or , the optimal bounds
are found by interpolating g and ,
as we have done. The optimal interpolation points
are shown to be the nodes of
a Gauss quadrature formula. With linear
bounds, this corresponds to interpolating
the value of g for and , or interpolating
the value and derivative of g for .

In [45], a collection of constrained
approximation problems is brought together,
with one-sided approximation treated as a special
case of general constrained approximation problems.
Linear programming is suggested as a method to
determine bounds when the nth derivative crosses zero:
[44] is cited. See [1, 43]
for more recent work. Much of the current discussion
is of spline aproximations, as in [43].
In all of the papers referenced, a detailed computational
procedure must be followed to determine an
approximate lower or upper bound.

In [72], another approach to proving
upper and lower polynomial bounds optimal is taken.
With this approach, it is shown that
interpolating the value and/or derivative
at the nodes of a Gauss quadrature formula
constructs the optimal polynomial
bound, provided that the bound does not
interpolate the function elsewhere.

Most of these results generalize to non-polynomial
bounds. Often, the bounds are taken from a
Chebyshev system [18, 45, 1, 44];
the set of n degree polynomials form a Chebyshev system.
In [72], bounds are taken as linear combinations
of an arbitrary
set of continuous functions.
Characterizations of constrained approximation
solutions has also been studied [30, 70].